Density functional theory study of small fe, co, ni and pt clusters on graphene

1595.11 Atomization energies, bond lengths and angles, and projected magneticmoments and electronic configurations for: a Free Fe2Co XZ orien-tation, b Free Fe2Co XY orientation, c Adsor

Density-Functional Theory study of small Fe, Co, Ni and Pt clusters on

December 29, 2008

Science is a differential equation.Religion is a boundary condition.– Alan Turing

Depart-2

Small metal clusters have properties that are distinct from the bulk and are thereforeinvestigated rather intensely for both fundamental and technological reasons In partic-ular much attention has been focused on the small clusters of the ferromagnetic metals,

Fe, Co and Ni, in the context of developing novel magnetic materials with high netization densities for use as storage media Due to the high surface-to-volume ratio,the electronic and magnetic properties, and therefore functionality, of these clusters areextremely sensitive to their immediate environment For the purposes of device appli-cation, these clusters have to be in the condensed phase and are either embedded within

mag-a mmag-atrix or mag-adsorbed on mag-a substrmag-ate The recent isolmag-ation of grmag-aphene hmag-as spmag-arked mmag-any

to investigate a myriad of possible applications given the rich physics associated withthis two-dimensional material As a substrate for these clusters, graphene might there-fore allow for an integration of technologies (e.g spintronics) In this work, I reportthe results of plane-wave density functional theory (DFT) calculations of the homonu-clear and heteronuclear Fe, Co, Ni and Pt adatoms, dimers, trimers and tetramers ad-sorbed on graphene All calculations in this work were performed using the Perdew-Burke-Ernzerhof (PBE) functional for the wavefunction with energy cutoffs of 40Ryand 480Ry for the wavefunction and density respectively Brillouin Zone sampling wasperformed with a Monkhorst-Pack grid of (8×8×1) There are two main aims in thiswork The first aim of this work involves investigating the suitability of graphene as asupport material for the small (up to the tetramer) homonuclear Fe, Co and Ni clusters.This suitability is determined by the extent to which the cluster-graphene interactionaffects the magnetic moment of these clusters relative to their respective gaseous states

3

Abstract 4The adsorption site configuration and relative stabilities, and the projected electronicconfigurations and magnetic moments of these clusters are studied The second aim

of this work involves investigating if enhanced binding and projected magnetic ments can be achieved by adsorbing the heteronuclear Fe, Co, Ni and Pt dimers, andselected heteronuclear trimers and tetramers on graphene The most stable dimer andtrimer configurations are those where the dimer bond axis and the trimer plane are ori-ented perpendicular to the graphene plane, and the most stable tetramer configuration

mo-is one where the tetramer mo-is adsorbed in the 3+1 configuration (i.e three atoms close

to graphene and one atom farther away) The total magnetic moments of the adsorbedhomo- and hetero-nuclear dimers are very similar compared to their respective gaseousstates On the other hand, the total magnetic moments of the adsorbed trimers andtetramers are reduced compared to their respective gaseous states Further to this, theprojected magnetic moments of adsorbed atoms close to the graphene plane are re-duced while the projected magnetic moments of the atoms farther from graphene areenhanced, both compared to their respective projected magnetic moments of the clus-ters in the gaseous state For the adsorbed heteronuclear dimers, the projected magneticmoments of Fe, Ni and Pt are most enhanced when bonded with Co The total mag-netic moments of the Fe-Pt and Co-Pt trimers and tetramers are enhanced relative tothe sum of the total magnetic moments of the homonuclear clusters that form them,while they are reduced in the cases of the Fe-Co and Ni-Pt trimers and tetramers Thestabilities of the adsorbed clusters are intricately dependent on the energy needed for

an electronic interconfigurational change that accompanies the desorption of these ters from graphene, geometry constraints (if any) and the amount of cluster-to-graphenecharge transfer The accuracy of the binding energies thus calculated would therefore

clus-be particularly dependent on how well the exchange-correlation functional used in thesecalculations treats the interconfigurational energy and the associated electronegativities

of the metals studied in this work Based on previous theoretical and experimentalwork, the magnetic moments calculated here are accurate

1.1 General introduction 20

1.2 Cluster studies: Gas phase 22

1.3 Cluster studies: Condensed phase 25

1.4 Graphene 28

1.5 Theoretical studies: accuracy and problems 31

1.6 Aims and organization of this work 32

2 Theoretical Foundations 35 2.1 The Schr¨odinger equation and Dirac notation 35

2.2 The Variational Principle 37

2.3 The Hellmann-Feynman Theorem 39

2.4 Hartree-Fock Theory 40

2.5 Density Matrices 46

2.5.1 Reduced density matrices 47

2.5.2 Spinless density matrices and the Dirac exchange functional 49

2.6 The Thomas-Fermi-Dirac Model 53

2.7 The Hohenberg-Kohn Theorems 56

2.8 The Kohn-Sham method 58

2.9 Exchange-Correlation: LDA and GGA 60

2.9.1 The Local Density Approximation 61 2.9.2 Gradient expansions and the Generalized Gradient Approximation 62

5

Contents 6

2.10 The art of Pseudopotentials 64

2.10.1 The existence of a pseudopotential 64

2.11 Reciprocal space and the plane wave basis 69

2.12 Fermi-Dirac statistics and Janak’s Theorem 73

2.13 Practical solution of the eigenvalue problem 76

3 Density functional theory study of Fe, Co and Ni adatoms and dimers on graphene 80 3.1 Introduction 80

3.2 Computational method and calibration 83

3.3 Results 86

3.3.1 Adatoms 86

3.3.2 Dimers 93

3.4 Conclusion 108

4 Adsorption Structures and Magnetic Moments of FeCo, FeNi, CoNi, FePt, CoPt and NiPt dimers on graphene 109 4.1 Introduction 109

4.2 Computational method 113

4.3 Results and Discussion 114

4.4 Conclusion 128

5 DFT study on the thermodynamics and magnetic properties of the homonu-clear Fe, Co and Ni trimers and tetramers, and selected heteronuhomonu-clear Fe, Co, Ni and Pt trimers and tetramers on graphene 130 5.1 Introduction 130

5.2 Computational method 133

5.3 Results and Discussion 133

5.3.1 Homonuclear Trimers 133

5.3.2 Homonuclear Tetramers 149

5.4 Mixed clusters 157

Contents 7

5.4.1 FeCo trimers and tetramers 160

5.4.2 FePt trimers and tetramers 165

5.4.3 CoPt trimers and tetramers 171

5.4.4 NiPt trimers and tetramers 176

5.5 Formation energies of the mixed clusters on graphene and changes in the magnetic moments 180

5.6 Conclusion 182

List of Figures

1.1 Representation of a zoomed-in image of the band structure of graphene,centered at the wavevector K at which the π (below the Fermi level orthe wavevector axis) and π∗ (above the Fermi level or the wavevectoraxis) bands just touch At this point, where the two bands just touch,the electrons behave relativistically and are referred to as Dirac fermions 302.1 Allowed wavevectors for a particle in a box 692.2 (a)The real space unit cell, represented by the red hexagon in the back-ground and labeled with vectors ~a1and ~a2, and the reciprocal space unitcell, also called the Brillouin zone, represented by the blue hexagon

in the foreground and labeled with the vectors ~b1 and ~b2, of graphene,(b) The Brillouin zone (shown in white) is the Wigner-Seitz cell of thereciprocal space lattice (shown in blue) The irreducible wedge of theBrillouin zone of graphene is shown in green and the high symmetrypoints are labeled asΓ, K and M 712.3 Plots of occupancy vs energy (eV) at four temperatures: 0K (blue line),

104K (green line), 105K (orange line), 106K (red line) 742.4 Flowchart of how a typical self-consistent field calculation is done usingdensity functional theory See text for details Diagram adapted fromRef [1] 78

8

List of figures 9

3.1 Convergence testing: 3.1(a): The total energies of the Fe, Co and Niatoms taken relative to the minimum total energy in each set as a func-tion of the lateral dimension of the supercell (in multiples of the unitcell of graphene, 2.46Å) using 40Ry and 480Ry for the wavefunctionand electron density cutoffs respectively 3.1(b): Dependence of thebinding energy of an Fe adatom adsorbed at a hole site on the wave-function and electron density cutoffs The electron density cutoff isgiven as F × (wavefunction cutoff) 3.1(c): Binding energy of an Feadatom as a function of the Monkhorst-Pack grid used Note that inall cases, a Marzari-Vanderbilt smearing width of 0.001Ry and a forceconvergence threshold of 0.001a.u were used 873.2 Schematic illustration of the adsorbed adatom configurations (top view):(a) Adatom above a hole site and (b) adatom atop a carbon atom orabove an atom site The hexagon represents the six nearest carbonatoms found in the graphene layer 883.3 The projected density of states for configurations 1.1 (3.3(a), 3.3(c),3.3(e)) and 1.2 (3.3(b), 3.3(d), 3.3(f)) for Fe, Co and Ni respectively.The Fermi level is referenced at 0eV Alpha and beta refer to the ma-jority (spin-up) and minority (spin-down) spin states respectively Thealpha and beta density of states overlap exactly in 3.3(e) and 3.3(f).The raising of both s spin states above the Fermi level in 3.3(a), 3.3(c),3.3(e) and 3.3(f) results in a decrease of 2µB for the magnetic moment

of Fe, Co and Ni when bound as configuration 1.1 (above a hole site)and of Ni when bound as configuration 1.2 (above an atom site) respec-tively Only the beta (minority or spin-down) s states are raised abovethe Fermi level in 3.3(b) and 3.3(d) which results in little change in themagnetic moment of Fe and Co when bound as configuration 1.2(above

an atom site) respectively Insets zoom in on the density of states within0.1eV of the Fermi level 92

List of figures 10

3.4 Band structure and density of states for graphene as calculated in thiswork 943.5 Bandstructures for Fe, Co and Ni (Figures 3.5(a), 3.5(b) and 3.5(c) re-spectively) when bound as configuration 1.1 (above a hole site), i.e themore stable adatom configuration The spin-bands overlap exactly inthe case of Ni 943.6 Representation of the dimer configurations (top view): Dimers above(a) 2 hole sites and (b) above 2 bridge sites with bond axes parallel tothe graphene plane and dimers above (c) a hole site and (d) above anatom site with bond axes perpendicular to the graphene plane Notethat the spheres in Figures (c) and (d) appear larger for the reason thatthe dimer is bound with its bond axis perpendicular to the graphene plane 953.7 Data for the bound dimers: [dimer species (dimer configuration)] (a)Fe(2.1), (b) Fe(2.2), (c) Fe(2.3), (d) Fe(2.4), (e) Co(2.1), (f) Co(2.2), (g)Co(2.3), (h) Co(2.4), (i) Ni(2.1), (j) Ni(2.2), (k) Ni(2.3), (l) Ni(2.4) Inset

in each subfigure’s top left corner is a top view of that configuration

as per shown in figure 3.6 (i.e in the x-y plane) The main figuregives the side view (i.e in the x-z plane) Shown in the figures arethe atomization energies (Eat), binding energies (Eb), the local charge

on each species, the local magnetic moments, the projected electronicconfiguration, the bound dimer’s bond length and the average metal-to-graphene separation Note that the baseline represents the grapheneplane and C is a symbol used to represent the whole graphene plane andnot just a single C atom found therein 1033.8 Electron density isosurface (isodensity value = 0.06 a.u.) for the var-ious bound Fe dimers The weakening of the Fe-Fe bond in config-uration 2.1 is well evidenced by the depreciation in electron densitybetween the two atoms relative to the other cases The pictures weregenerated using XCrysden[2] 107

List of figures 11

4.1 Dissociation energies (Ed), bond lengths, projected magnetic momentsand electronic configurations of the free FeCo, FeNi, CoNi, FePt, CoPtand NiPt dimers The color code for Fe, Co, Ni and Pt is red, green,purple and gray respectively and will be used throughout this chapter

We note that the charges do not balance exactly and is a result of theerrors introduced when calculating and integrating the projected density

of states 1164.2 Representations of the four general initial configurations of the six mixeddimers studied in this work There are two sub-configurations of thetype 2.3 and 2.4 which we have called 2.3.1 and 2.3.2, and 2.4.1 and2.4.2, where the lower index corresponds to the case where the specieswith the higher proton number is closer to graphene Not all initialconfigurations are stable This is discussed in the text 1184.3 The atomization (Eat) and binding (Eb) energies, metal-metal bond lengths,metal-to-graphene separation, and projected magnetic moments and elec-tronic configurations of the bound FeCo dimers Configuration 2.4.1 isunstable and the dimer with that initial configuration converged to con-figuration 2.3.1 1194.4 The atomization (Eat) and binding (Eb) energies, metal-metal bond lengths,metal-to-graphene separation, and projected magnetic moments and elec-tronic configurations of the bound FeNi dimers Configurations 2.4.1and 2.4.2 are unstable and the dimers with those initial configurationsconverged to configurations 2.3.1 and 2.3.2 respectively 1204.5 The atomization (Eat) and binding (Eb) energies, metal-metal bond lengths,metal-to-graphene separation, and projected magnetic moments and elec-tronic configurations of the bound CoNi dimers Configurations 2.1,2.4.1 and 2.4.2 are unstable and the dimers with those initial configura-tions converged to configurations 2.2, 2.3.1 and 2.3.2 respectively 121

List of figures 12

4.6 The atomization (Eat) and binding (Eb) energies, metal-metal bond lengths,metal-to-graphene separation, and projected magnetic moments and elec-tronic configurations of the bound FePt dimers Configuration 2.3.1

is unstable and the dimer with that initial starting configuration verged to the configuration labeled 2.3.1.0, with the Pt atom (closer tographene) located above the bridge site (i.e at the mid-point of theC-C bond in graphene) Configurations 2.1, 2.2 and 2.4.2 are unstableand the dimers with those initial configurations all converged to con-figuration 2.3.2 respectively, albeit the latter being the least stable ofthe bound FePt configurations studied in this work suggesting that theenergy barrier to configuration 2.3.2 is lower than the energy barrier toconfiguration 2.3.1.0, the global minimum of the configurations studiedhere 1234.7 The atomization (Eat) and binding (Eb) energies, metal-metal bond lengths,metal-to-graphene separation, and projected magnetic moments and elec-tronic configurations of the bound CoPt dimers Like the FePt dimerbound with an initial configuration 2.3.1, the CoPt dimer converged toconfiguration 2.3.1.0 Configurations 2.1 and 2.2 are unstable and thedimers with those initial configurations both converged to configura-tion 2.3.2, which in the case of the bound CoPt, is the most stable of thebound CoPt dimer configurations studied in this work 1244.8 The atomization (Eat) and binding (Eb) energies, metal-metal bond lengths,metal-to-graphene separation, and projected magnetic moments and elec-tronic configurations of the bound CoPt dimers Like the FePt and CoPtdimers bound with an initial configuration 2.3.1, the NiPt dimer con-verged to configuration 2.3.1.0 Configuration 2.1 is unstable and thedimer with that initial configuration converged to configuration 2.2 125

con-List of figures 13

5.1 Atomization energies, bond angles, bond lengths and projected netic moments and electronic configurations of the free Fe, Co and Nitrimers Two stable Fe trimer geometries were obtained by changingthe orientation of this trimer in the supercell, which is anisotropic, used

mag-in our calculations: an isosceles triangle and an equilateral triangle Forthe Co and Ni trimers, the same bond lengths, bond angles and pro-jected electronic configurations and magnetic moments were obtainedregardless of the orientation of these trimers within the supercell 1405.2 Representations of the six adsorbed trimer configurations studied in thiswork The spheres represent the metal atoms Larger spheres indicatethat those metal atoms are further from the graphene plane relative tothe smaller spheres: (a) each adatom above a hole site, (b) each adatomatop an atom site, (c) two adatoms above hole sites and a third atom,further from graphene, above the bridge site, (d) two adatoms abovebridge sites with a third atom, further from graphene, above the holesite, (e) one adatom above the bridge site with two atoms, further fromgraphene above neighbouring hole sites and (f) one adatom above thehole site with two atoms, further from graphene above neighbouringbridge sites 1415.3 Geometric (bond lengths and angles) and geometry related data (lo-cal charges and magnetic moments) of the various adsorbed Fe trimerconfigurations on graphene The subfigure captions specify the con-figuration of interest The insets in subfigures (a) and (b) represent aside-profile view (in the xz plane), where the single sphere representsthe whole trimer Subfigures (c)-(f) illustrate the trimer as viewed inthe xz plane The top view of all conformers are shown at the top-leftcorner of each subfigure 146

List of figures 14

5.4 Geometric (bond lengths and angles) and geometry related data cal charges and magnetic moments) of the various adsorbed Co trimerconfigurations on graphene The subfigure captions specify the con-figuration of interest The insets in subfigures (a) and (b) represent aside-profile view (in the xz plane), where the single sphere representsthe whole trimer Subfigures (c)-(f) illustrate the trimer as viewed inthe xz plane The top view of all conformers are shown at the top-leftcorner of each subfigure 1475.5 Geometric (bond lengths and angles) and geometry related data (lo-cal charges and magnetic moments) of the various adsorbed Ni trimerconfigurations on graphene The subfigure captions specify the con-figuration of interest The insets in subfigures (a) and (b) represent aside-profile view (in the xz plane), where the single sphere representsthe whole trimer Subfigures (c)-(f) illustrate the trimer as viewed inthe xz plane The top view of all conformers are shown at the top-leftcorner of each subfigure 1485.6 Atomization energies (Eat), bond lengths, and projected magnetic mo-ments, and electronic configurations of the free Fe, Co and Ni tetramers.Two configurations, based on the anisotropy of the supercell used in thecalculations in this work, were studied: a 2+2 configuration and a 3+1configuration (see text for details of these configurations) 1525.7 Representations of the four initial configurations of the adsorbed homonu-clear tetramers The spheres represent the metal atoms Larger spheresindicate that those metal atoms are further from the graphene plane rel-ative to the smaller spheres: (a) each adatom above a hole site, (b) eachadatom atop an atom site, (c) three adatoms above hole sites and a thirdatom, further from graphene, above the atom site, (d) three adatomsabove atom sites with a third atom, further from graphene, above thehole site 153

(lo-List of figures 15

5.8 Geometric (bond lengths and angles) and geometry related data cal charges and magnetic moments) of the various Fe3 conformers ongraphene The subfigure captions specify the conformer of interest Theinsets in subfigures (a) and (b) represent a side-profile view (in the xzplane), where the single sphere represents the whole trimer Subfigures(c)-(f) illustrate the trimer as view in the xz plane The top view of allconformers are shown at the top-left corner of each subfigure 1575.9 Geometric (bond lengths and angles) and geometry related data (lo-cal charges and magnetic moments) of the various Co3 conformers ongraphene The subfigure captions specify the conformer of interest Theinsets in subfigures (a) and (b) represent a side-profile view (in the xzplane), where the single sphere represents the whole trimer Subfigures(c)-(f) illustrate the trimer as view in the xz plane The top view of allconformers are shown at the top-left corner of each subfigure 1585.10 Geometric (bond lengths and angles) and geometry related data (lo-cal charges and magnetic moments) of the various Ni3 conformers ongraphene The subfigure captions specify the conformer of interest Theinsets in subfigures (a) and (b) represent a side-profile view (in the xzplane), where the single sphere represents the whole trimer Subfigures(c)-(f) illustrate the trimer as view in the xz plane The top view of allconformers are shown at the top-left corner of each subfigure 1595.11 Atomization energies, bond lengths and angles, and projected magneticmoments and electronic configurations for: (a) Free Fe2Co (XZ orien-tation), (b) Free Fe2Co (XY orientation), (c) Adsorbed Fe2Co (configu-ration 3.3) 1615.12 Atomization energies, bond lengths and angles, and projected magneticmoments and electronic configurations for: (a) Free FeCo2 (XZ orien-tation), (b) Free FeCo2(XY orientation), (c) Adsorbed FeCo2(configu-ration 3.3) 162

(lo-List of figures 16

5.13 Atomization energies, bond lengths and angles, and projected magneticmoments and electronic configurations for: (a) Free Fe3Co (2+2 config-uration), (b) Free Fe3Co (3+1 orientation), (c) Adsorbed Fe3Co (config-uration 4.3) 1635.14 Atomization energies, bond lengths and angles, and projected magneticmoments and electronic configurations for: (a) Free FeCo3(2+2 config-uration), (b) Free FeCo3(3+1 orientation), (c) Adsorbed FeCo3(config-uration 4.3) 1645.15 Atomization energies, bond lengths and angles, and projected magneticmoments and electronic configurations for: (a) Free Fe2Pt (XZ orienta-tion), (b) Free Fe2Pt (XY orientation), (c) Adsorbed Fe2Pt (configura-tion 3.3) 1665.16 Atomization energies, bond lengths and angles, and projected magneticmoments and electronic configurations for: (a) Free FePt2(XZ orienta-tion), (b) Free FePt2 (XY orientation), (c) Adsorbed FePt2 (configura-tion 3.4) 1675.17 Atomization energies, bond lengths and angles, and projected magneticmoments and electronic configurations for: (a) Free Fe3Pt (2+2 config-uration), (b) Free Fe3Pt (3+1 orientation), (c) Adsorbed Fe3Pt (configu-ration 4.3) 1685.18 Atomization energies, bond lengths and angles, and projected magneticmoments and electronic configurations for: (a) Free FePt3(2+2 config-uration), (b) Free FePt3(3+1 orientation), (c) Adsorbed FePt3(configu-ration 4.4) 1695.19 Atomization energies, bond lengths and angles, and projected magneticmoments and electronic configurations for: (a) Free Co2Pt (XZ orienta-tion), (b) Free Co2Pt (XY orientation), (c) Adsorbed Co2Pt (configura-tion 3.3) 171

List of figures 17

5.20 Atomization energies, bond lengths and angles, and projected magneticmoments and electronic configurations for: (a) Free CoPt2(XZ orienta-tion), (b) Free CoPt2(XY orientation), (c) Adsorbed CoPt2 (configura-tion 3.3) 1725.21 Atomization energies, bond lengths and angles, and projected magneticmoments and electronic configurations for: (a) Free Co3Pt (2+2 orien-tation), (b) Free Co3Pt (3+1 orientation), (c) Adsorbed Co3Pt (configu-ration 4.4) 1735.22 Atomization energies, bond lengths and angles, and projected magneticmoments and electronic configurations for: (a) Free CoPt3(2+2 config-uration), (b) Free CoPt3(3+1 orientation), (c) Adsorbed CoPt3(config-uration 4.4) 1745.23 Atomization energies, bond lengths and angles, and projected magneticmoments and electronic configurations for: (a) Free Ni2Pt (XZ orienta-tion), (b) Free Ni2Pt (XY orientation), (c) Adsorbed Ni2Pt (configura-tion 3.4) 1765.24 Atomization energies, bond lengths and angles, and projected magneticmoments and electronic configurations for: (a) Free NiPt2(XZ orienta-tion), (b) Free NiPt2 (XY orientation), (c) Adsorbed NiPt2 (configura-tion 3.4) 1775.25 Atomization energies, bond lengths and angles, and projected magneticmoments and electronic configurations for: (a) Free Ni3Pt (2+2 orien-tation), (b) Free Ni3Pt (3+1 orientation), (c) Adsorbed Ni3Pt (configu-ration 4.4) 1785.26 Atomization energies, bond lengths and angles, and projected magneticmoments and electronic configurations for: (a) Free NiPt3 (2+2 orien-tation), (b) Free NiPt3 (3+1 orientation), (c) Adsorbed NiPt3 (configu-ration 4.4) 179

List of Tables

3.1 The spin-orbital occupancies for the free atom 853.2 Data for Fe, Co & Ni adatom adsorption on graphene: The bindingenergies (Eb), the magnetic moments (M), the metal-to-graphene planedistance (L), the metal-to-graphene charge transfer in units of electrons(q) and the electronic configuration of the metal atoms when bound inthe respective configurations 913.3 Local magnetic moments for the adatoms and the graphene 933.4 Binding energy, magnetic moment and bond length of the free Fe dimer 963.5 Binding energy, magnetic moment and bond length of the free Co dimer 973.6 Binding energy, magnetic moment and bond length of the free Ni dimer 983.7 Comparison of data for configurations 2.1 and 2.2 with the work ofDuffy & Blackman and Yagi et al 1013.8 Percentage change in the bound dimers’ bond lengths with respect totheir respective unbound cases 1043.9 The relative binding strength for each of the bound dimer configurationsrelative to having 2 adatoms adsorbed at their respective most stablesite(i.e configuration 1.1 - adatom above a hole site) for all of Fe, Coand Ni 1074.1 The spin-orbital occupancies derived from each atoms’ projected den-sity of states 115

18

List of tables 19

5.1 The spin-orbital occupancies derived from each atoms’ projected sity of states 1345.2 Dissociation energies (Ed), total magnetic moments (TMM), bond lengths(R) and bond angles (θ) of the free Fe trimer 1355.3 Dissociation energies (Ed), total magnetic moments (TMM), bond lengths(R) and bond angles (θ) of the free Co trimer 1375.4 Dissociation energies (Ed), total magnetic moments (TMM), bond lengths(R) and bond angles (θ) of the free Ni trimer 1385.5 The formation energies (eV) of each adsorbed trimer configuration, foreach homonuclear metal species, as a result of the reaction of a dimerand an adatom (see Equation 5.1) In general, Fe shows the greatesttendency to agglomeration, while Ni shows the least 1495.6 The formation energies (eV) of each adsorbed trimer configuration, foreach homonuclear metal species, as a result of the reaction of threeadatoms (see Equation 5.2) Again, Fe shows the greatest tendency toagglomeration, while Ni shows the least 1495.7 The formation energies (eV) of each adsorbed tetramer configuration,for each homonuclear metal species, as a result of the reaction of atrimer and an adatom (see Equation 5.3) In general, Fe shows the great-est tendency to agglomeration, while Ni shows the least 1565.8 The formation energies (eV) of each adsorbed tetramer configuration,for each homonuclear metal species, as a result of the reaction of twodimers (see Equation 5.4) 1565.9 Formation energies (in eV) and change in the total absolute magneticmoments, given in brackets (in µB), for the respective heteronucleartrimer (Equations 5.5 and 5.6) and tetramer (Equations 5.7 and 5.8)formation reactions 182

technolog-in the bulk phase The breaktechnolog-ing of bond symmetry at the surface results technolog-in a substantialmodification of the electronic structure of the cluster compared to the bulk phase andoften leads to an enhancement in several physical and chemical properties which aredesired and advantaged in novel device application The physical properties of theseclusters do not vary monotonically with cluster size and offers the intriguing possibility

to use clusters of various sizes for a variety of technologies This also presents a damental challenge, both experimentally and theoretically, in context of studying andcharacterizing the ground state properties of these clusters

fun-Two main applications of transition metal clusters are in the fields of catalysis andadvanced magnetic materials For example, Ni, Pt and bimetallic Ni/Pt clusters havebeen investigated as alternatives to oxygen electroreduction catalysts for the develop-ment and improvement of fuel cell technology [3, 4, 5] Fe clusters and Fe-COnclusters

20

1.1 General Introduction 21

have attracted much attention as catalysts for the growth of single- and multi-walled bon nanotubes [6, 7, 8, 9] In context of advanced magnetic materials, transition metalclusters and nanocomposites, particularly those of and/or containing the ferromagneticmetals Fe, Co and Ni, have found particular use in the design of magnetic materialsfor use in magneto-optical recording, magnetic sensors, high-density magnetic mem-ory, optically transparent materials, soft ferrites, nanocomposite magnets, spintronicdevices, magnetic refrigerants, high-TC superconductors, ferrofluids and for biologicalapplications (e.g in cancer thermal ablation treatment) [10] Of particular interest inrecent years is the design of magnetic materials with large magnetic anisotropies, highcoercivities and high magnetization densities for use as magnetic media in high-densitymagnetic data storage To surpass the 10 Gbit in−2 density limit in the areal density inlongitudinal recording, a reduction in grain or cluster size and control of inter-grain ex-change coupling is desired [10] However, an associated problem with decreasing grain

car-or cluster size is the fact that these clusters become superparamagnetic This means thatthe clusters, should they have low blocking temperatures (Curie or Neel), do not retaintheir magnetization upon the turning off of the externally applied magnetic field Thisposes a problem in context of their use and much research effort is spent on studyingand understanding the fundamental physics of these clusters

The transition metal clusters are studied both experimentally and theoretically inboth the gas and condensed phases There are two main classes in context of the latter:adsorbed clusters on various substrates and clusters embedded in a matrix of anothermaterial Due to the reduction of bond density, particularly at the surface, these clustersare extremely sensitive to their environments Therefore, we expect that the substrates

or matrices that support or embed the clusters are of considerable importance in text of achieving specific electronic and magnetic properties (e.g high magnetizationdensities) and are therefore investigated rather intensely Many materials have been in-vestigated, some proving to be particularly suited for magnetic application Less than 5years ago, a ‘new’ material was isolated Since then, this material has not failed to sur-prise the scientific community with its rich physics and possible myriad applications

con-1.2 Cluster studies: Gas phase 22

Nothing more than two-dimensional array of carbon atoms arrange in a honey-comblattice, graphene has taken the fore in materials research A natural question to then ask

is if graphene might be a suitable substrate, particularly for transition metal clusters tocarry out both catalytic and magnetic work Aside from the functionality of the tran-sition metal clusters themselves, their presence could sufficiently alter the electronicproperties of graphene itself and might thereby allow for an integration of technologies

In this chapter, I will review some of the work that has been done in investigating thesmall clusters of the paramagnetic metals, Fe, Co and Ni These include the homonu-clear and heteronuclear clusters containing these elements which have been studied inboth the gas and condensed phases As I am interested in the interaction of these clus-ters with graphene, I will give a review of this recently isolated allotrope of carbon andsome of its interesting properties, which in itself motivates this study Given the sizeregime that is studied in this work, theoretical calculations have to be carried out in or-der to get a handle of the phenomena that occur at the atomic and even electronic scale

In particular, density functional theory calculations are used It is therefore importantthat I point out the limitations associated with such calculations in context of the systemstudied in this work Finally, I will end this chapter by outlining the structure of thisthesis, viz the particular questions that I wish to address and how I resolve to answerthem

1.2 Cluster studies: Gas phase

Bulk Fe, Co and Ni are known to be ferromagnetic with magnetization values of2.25, 1.67 and 0.67 µB/atom respectively Although most elements (except the noblegases) are paramagnetic in their atomic or small cluster state, these ferromagnetic met-als are suitable materials for the development of novel magnetic materials since theyoffer the potential of having amongst the largest magnetization values as small clusters.For example, theoretical calculations have concluded that gaseous Fe2, Co2 and Ni2dimers have magnetization values 3µ /atom [11, 12, 13, 14, 15, 16], 2µ /atom [12, 17]

1.2 Cluster studies: Gas phase 23

and 1µB/atom [12, 18, 19, 20, 21, 22, 23] respectively In the case of the trimers, though higher-than-bulk magnetization values are predicted in general, there is littleconsensus in context of determining the ground-state electronic state, and therefore theground-state magnetization value For the Fe trimer, Papas et al [24], and Gutsev et al[25], calculated an average magnetic moment of 3.33µB/atom while K¨ohler et al [14]and Castro et al [11, 12] calculated an average magnetic moment of 2.67µB/atom Forthe Co trimer, Fan et al [26] calculated an average magnetic moment of 2.33µB/atomwhile Papas et al [24] and Jamorski et al [12, 17] calculated an average magnetic mo-ment of 1.67µB/atom For the Ni trimer, most calculations [12, 20, 21] conclude that theaverage magnetic moment is 0.67µB/atom Conclusively determining the ground-stategeometry and electronic state for the small Fe, Co and Ni clusters is still a fundamentalproblem and one that is extremely difficult to resolve theoretically given the limits ofthe current levels of theory employed in most calculations These limitations, which

al-I refer to in explaining some of the results that al-I have obtained in this work, will beelaborated on later in this chapter

With increasing cluster size, the average magnetic moment decreases albeit monotonically As the average bond density of the cluster tends towards that of thebond density found in the bulk, a sufficiently large cluster would have the same magne-tization value as that of the bulk phase It has been also shown that for certain clustersizes, or clusters with a “magic number” of atoms, the magnetization value can be muchhigher and lower than expected These “jumps” in an already non-monotonic trend areinteresting not just from an application point of view, but also in context of under-standing the electronic structure of clusters and how these evolve as a function of thecluster size The phase space of the small homonuclear clusters, even in considering the

non-“magic clusters”, is still limited in context of pushing the boundaries of higher zation values The Slater-Pauling curve shows that bulk alloying can result in enhancedand reduced magnetization values depending on the species involved and the respectivestoichiometries employed In similar vein, heteronuclear clusters may be the key inunlocking the door behind which lies materials with enhanced chemical and physical

magneti-1.3 Cluster studies: Condensed phase 24

properties

Little work has been done to investigate the ground-state properties of the clear clusters of Fe, Co and Ni (e.g FeCo, FeNi, CoNi) An important piece of work incontext of assigning the ground state properties of heteronuclear dimers was carried out

heteronu-by Gutsev et al [27] They calculated the ground-state bond lengths, projected tronic configurations and magnetic moments, vibrational frequencies and ionization po-tentials for the mixed 3d-metal dimers using density functional theory, with a 6-311+G*basis set and the BPW91 [28, 29] exchange-correlation functional We would expectthat for 3d transition metal atoms with less than five d-state electrons, the magnetic mo-ment for the more electronegative species involved in the dimer pair would be enhancedwhile that of the less electronegative species would be reduced On the other hand, for3d transition metal atoms with five or more d-state electrons, the magnetic moment forthe more electronegative species would be reduced while that of the less electronegativespecies would be enhanced Gutsev et al found this to be the case and they establishedthe extent to which magnetic moment enhancement and reduction occurs for a variety

elec-of mixed dimers

Andriotis et al [30, 31] went a step further in trying to understand the basis formagnetic enhancement and reduction in a small magnetic cluster, and the disagreementbetween experimentally measured and theoretically calculated magnetic moments ofthese clusters They found, using tight-binding molecular dynamics calculations, that it

is not always the case that high orbital states are energetically favored; the true state of the cluster of interest would therefore be lower than expected Allowing thesehigh orbital states to be energetically favored would involve cluster-substrate interactionwhich would then give rise to higher overall magnetic moments Apart from the abovementioned studies, little else has been done to investigate small heteronuclear clusters

ground-in the gas phase, let alone their ground-interactions with various substrate materials

1.3 Cluster studies: Condensed phase 25

1.3 Cluster studies: Condensed phase

Clusters are of little use when they are in the gas phase Potential application ofthese clusters requires them to be in the condensed phase: either adsorbed on a sub-strate or embedded within a matrix As the bond density of these cluster is considerablyreduced relative to its bulk phase, these clusters are extremely sensitive to their imme-diate environment which could potentially alter the electronic state of these clusters.Significant substrate-cluster interactions may or may not be advantageous: magneticmoments may be enhanced in some cases but reduced in other cases A large variety ofstudies have been therefore done to investigate how the Fe, Co and Ni clusters interactwith a host of materials

Several studies have been done to investigate the interaction, and effect on clustermagnetization that metal substrates have on the Fe, Co and Ni clusters Lazarovits et

al [32] studied the small Fe, Co and Ni clusters on the Ag(100) surface They foundthat for the Fe and Co adatoms, the overall magnetic moment is enhanced relative to thefree atom state, a large proportion of this enhancement being due to orbital magneticenhancement For the Ni adatom however, they found that the magnetic moment wascompletely quenched In context of magnetization anisotropies, they showed that the

Fe and Co adatoms have a strong tendency to perpendicular magnetism On the Feclusters retained this property; the Co, and Ni, clusters preferred a magnetic orientationthat was in-plane with the cluster itself Stepanyuk et al [33] calculated the magneticmoments of the 3d, 4d and 5d transition metal atoms adsorbed on Pd(001) and Pt(001).They found that compared to being adsorbed on Cu(001) and Ag(001), the magneticmoment for the Fe, Co and Ni adatoms is most enhanced when adsorbed on Pd(001).The magnetic moments on these adatoms do not change significantly when adsorbed

on Pt(001) as compared to when adsorbed on Pd(001) These studies indicate that theelectron affinity and ionization potential of the substrate, as well as the extent to which

it can induce spin-orbit interaction with the cluster, are critical in context of enhancingthe magnetization of the adsorbed cluster

Carbon nanotubes can be used to “tune” the magnetic moment of an adsorbed cluster

1.3 Cluster studies: Condensed phase 26

based on their curvature As mentioned, the electron affinity and ionization potential arecritical in context of determining the magnetic moment of an adsorbed cluster Just asthe metallicity of the carbon nanotube varies with its curvature, so too does its electronaffinity and ionization potential Menon et al [34] showed, using tight-binding, spin-unrestricted molecular dynamics calculations, that electron transfer from an adsorbedcluster to substrate occurs for carbon nanotubes with high curvature (i.e nanotubeswith small radii) As the curvature of the nanotube decreases (i.e as it tends towards

a more flat, graphene-like surface), this electron transfer process reverses Depending

on the species adsorbed, the electron transfer process can either enhance or reduce themagnetization of the cluster For example, they found that the magnetic moment of

Ni2 adsorbed on graphite is higher than in the case where the dimer is adsorbed on acarbon nanotube since the latter transfers charge to the cluster while the former drawscharge away from the cluster Some four years later, Yagi et al [35] further validatedthis point by calculating the adsorption site binding energies and magnetic momentsfor the Fe, Co and Ni adatoms and dimers adsorbed on a (4,4) carbon nanotube Theytoo found that for certain adsorption sites on this nanotube, the magnetic moments ofthe Fe, Co and Ni dimers are enhanced compared to the respective magnetic momentswhen adsorbed on graphene Doped carbon nanotubes might therefore be particularlyuseful for magnetic device and spintronics application In recent years, it is the buildingblock of all carbon allotropes, including that of carbon nanotubes, that has attractedmuch attention in many areas of research, including that of support materials research.Graphene, a single layer of graphite, has been shown to be suitable as a supportmaterial for Fe, Co and Ni adatoms and dimers Duffy and Blackman [36] and Yagi et

al [35] investigated the binding site adsorption energies and magnetic moments for the

Fe, Co and Ni adatoms and dimers adsorbed on graphene Both groups found that themagnetic moment of the adatom when adsorbed above a hole site (i.e in the middle

of a hexagon of carbon atoms; more will be said about this in Chapter 3) decreases by2µB When adsorbed above a atom site (i.e directly atop a carbon atom in graphene),the Fe and Co adatoms have magnetic moments of 4µ and 3µ respectively, while the

1.3 Cluster studies: Condensed phase 27

magnetic moment of the Ni adatom at the same site is still reduced by 2µB They alsofound that the Fe and Co dimers have magnetic moments that are the same as theirrespective gas phase moments even when adsorbed on graphene The main difference

in their work lies in the calculated values for the binding energies which is a result

of different computational methodologies being employed for each calculation Chan,Neaton and Cohen [37] also studied the adsorption site binding energy for the Fe adatom

on graphene and found an energy that was lower than that calculated by both Duffy andBlackman, and Yagi et al Mao et al [38] also calculated the binding site adsorptionenergies for the Fe and Co adatoms on graphene, and they too found energies thatdiffer from the previous mentioned groups The lack of conclusiveness in context ofdetermining the binding energies for these systems is a fundamental problem and stemsprimarily from the level of approximation used in these calculations The other majorfactor that contributes to this inconsistency is the different calculation parameters used.Nonetheless, all of these groups are in agreement in terms of the calculated values ofthe magnetic moments of these clusters when adsorbed on graphene

Clusters embedded within a matrix of another material have been shown to exhibitenhanced and reduced magnetic moments depending on the cluster species and matrixmaterial Given that the maximum of the Slater-Pauling curve [39] occurs for bulk

Fe0.7Co0.3, a natural starting point in designing a material with enhanced magnetizationdensity would be to consider small Fe clusters embedded within a Co matrix or viceversa In 2002, Xie and Blackman [40] showed using tight-binding calculations that themagnetic moment of Fe clusters within a size range of 100-600 atoms within a Co ma-trix is comparable to the magnetic moment of the free clusters They attributed this tothe effective exchange splitting on Fe due to the presence of the surrounding Co atoms.Interestingly enough, they found that the moment on these Fe clusters was reducedwhen surrounded by a Cu matrix In 2004, Bergman et al [41] further investigatedthis phenomenon by increasing the cluster size range to 700 atoms, and also probedthe projected magnetic moments of the Fe and Co atoms at the Fe/Co interface, as well

as the Fe and Co atoms that were farther form that interface Their density functional

1.4 Graphene 28

theory calculations showed that it is the Fe atoms at the interface that have their netic moments enhanced, while those embedded deep within the Fe cluster itself havemoments that are closer to the bulk magnetization value of Fe; the Co atoms displayedbulk magnetization values regardless of whether the atoms were at the interface or not

mag-On the same topic, Mpourmpakis et al [42] showed in 2005 that the enhanced magneticmoments on the Fe atoms stem from structural changes induced by the Co atoms Theseresults would point us to consider three points in context of how the magnetic moments

on clusters are enhanced or reduced First, the extent to which the cluster couples withthe surround material will have a bearing on the geometry of the cluster and thereforeits localized electronic state and magnetic moment Second, heteronuclear clusters con-taining Co in particular might exhibit enhanced magnetic moments Third, cluster size

as well as its coverage in or on the substrate are important in context of cluster-clusterinteractions and therefore magnetic coupling

be grown epitaxially, which would be more suited for mass production [44] Graphenecan be regarded as the basic building block of all graphitic forms, but there is muchmore to graphene than just its structural elegance

Ever since its isolation in 2004, there have been numerous publications on grapheneand its rich physics Novoselov et al [43] found electron and hole concentrations of

up to 1013 per square centimeter and room temperature mobilities of approximately

10000 square centimeters per volt-second can be induced by applying a gate voltageacross graphene, which is approximately ten times higher than the mobilities observed

1.4 Graphene 29

in commercial silicon wafers [45] Higher mobilities, which also implies higher drift locities, would allow for the development of more efficient electronic devices Anotherinteresting phenomenon associated with graphene is that one can observe the quantumHall effect even at room temperature In the classical Hall effect, an electric field, alsocalled the Hall field, is developed across two faces of a conductor when a current flowsacross a magnetic field [46] This Hall potential scales linearly with the current thatflows and the applied magnetic field In the quantum Hall effect however, the Hall po-tential is quantized When electrons are subjected to a magnetic field, they take the path

ve-of least resistance and maintain circular cyclotron orbits, with which is associated acyclotron frequency Each frequency or state is called a Landau Level (similar to the or-bitals that are used to describe the electronic structure of an atom) When the magneticfield is sufficiently strong and the electrons in the system occupy just a few Landau lev-els, the quantum Hall effect is observed To achieve the latter would nominally requirelow operating temperatures since this effect is dependent upon the effective mass of theelectron, which itself is a property that is dependent on temperature Graphene on theother hand exhibits the quantum Hall phenomenon even at room temperatures Heer-she et al also found that graphene can be used in making a superconducting transistor[47] Graphene itself is not superconducting However, it can relay a superconduct-ing current because it does not destroy the quantum coherence between the electronsthat form the Cooper pairs that are intrinsic to a superconducting current Therefore,what Heershe et al did was to bridge two superconducting electrodes with a film ofgraphene, thus creating a Josephson junction At the interface between the graphenesheet and the superconducting electrode, Andreev reflections occurs, in which an inci-dent electron (hole) is retroreflected as a hole (electron) of opposite spin and momentum

to the incident particle, thereby creating a Cooper pair in the superconductor [47] Suchdevices are particularly useful as magnetometers, single-electron transistors and are abasic component of quantum computers

The above novel phenomena, and others that are continually being discovered, areattributed to the two-dimensionality of graphene and the resulting peculiar nature of its

1.4 Graphene 30

electrons The band structure of graphene, shown in Figure 1.1, shows that it is a gap semiconductor or a semi-metal [46] In a usual semiconductor, there is a finite gapbetween valence and conducting bands, while in a metal, the valence and conductingbands overlap In the case of graphene, these bands just touch each other at points inmomentum- or Fourier- space known as Dirac points At these points, the electronsbehave as relativistic quasiparticles called Dirac fermions [48] Dirac fermions move atvelocities that are independent of their energy and direction, similar to photons albeitbeing fermions rather than bosons As such, the electrons in graphene behave as thoughthey were massless and thus gives rise to the above mentioned phenomena [45]

zero-Figure 1.1: Representation of a zoomed-in image of the band structure of graphene,centered at the wavevector K at which the π (below the Fermi level or the wavevec-tor axis) and π∗ (above the Fermi level or the wavevector axis) bands just touch Atthis point, where the two bands just touch, the electrons behave relativistically and arereferred to as Dirac fermions

An important question to then ask is if graphene can be coupled with other als such that novel technologies may arise or be integrated One aspect that has beenexplored thus far is the breaking of symmetry of the Bloch waves in graphene Thissymmetry breaking can be done by making use of an underlying support such as SiC[49], by introducing adsorbates onto graphene (e.g H adsorption on graphene [50]) or

materi-by introducing defects in graphene (e.g Stoner-Wales defect [50]) The most tant outcome of the above is that the band structure symmetry of graphene is brokenand does allow for the possibility of engineering band gaps that may find use in theelectronics industry More importantly, if a magnetic moment should be induced as a

impor-1.5 Theoretical studies: accuracy and problems 31

result of breaking the symmetry of the band structure, the different band gap energiesassociated with each spin may be advantaged in spintronic based devices such as thegeneration of spin-polarized current for use in mass-storage devices, spin-valves and inLEDs

1.5 Theoretical studies: accuracy and problems

Experimental methods are severely limited in context of studying various physicalproperties at the atomic scale With advanced hardware, more sophisticated algorithmsand ever-improving theoretical methods, computational modeling is almost necessarywhen it comes to studying materials at the nanometer scale In fact, experimentalistsoften couple or back their findings with theoretical calculations There are a host of the-oretical methods available today and the choice depends on what issues or questions are

to be answered In dealing with problems to do with geometry, electronic structure andmagnetism, density functional theory is perhaps the most efficient and computationallyeconomical tool available to the scientist today

There are quite a few limitations associated with density functional theory lations, some of which are particular pertinent to this work I will elaborate on thesehere As a theory, DFT is perfect and complete; all aspects of the fermionic interac-tions associated with electrons are succinctly contained within its equations (which Iwill elaborate later in this thesis) One such aspect is the electron exchange and cor-relation interactions and this is where DFT begins to falter While the equations inDFT explicitly state that the exchange-correlation energy can be expressed as a func-tional of the electron density, the problem lies in establishing the functional form forthese interactions, which today we only have one approximation after another Forcertain systems, especially where the electronic energy states have energies that aresufficiently wide apart, this is not a pressing problem When dealing with transitionmetals however, where the 3d and 4s energy levels are close, accurate treatment of theexchange-correlation interactions is paramount in establishing any conclusions derived

calcu-1.6 Aims and organization of this work 32

from the theoretical calculation If the level of approximation used in determining theform of the exchange-correlational functional is inaccurate, the true ground state prop-erties, viz geometry, electronic configuration, magnetic moments, dipole moments, cannever be commensurate with experiment

There are two main approaches to approximating the exchange-correlation tional: the local density approach, which is dependent only on the electron density atany given point in space, and the gradient-corrected approach, which is dependent notjust on the electron density at some point in space, but also on the change in electrondensity, the curvature of the electron density function at that point in space and so on(i.e higher orders in the Taylor series expansion) Calculations that make use of the lo-cal density approximation are usually notorious for poorly estimating binding energiesand electronic interconfigurational energies; gradient corrected exchange-correlationfunctionals take non-locality into account and this is particularly important in context

func-of chemical bonding [51, 52] One would then expect that the latter would be a betterchoice for almost all types calculations since non-locality is taken into account This isnot true, especially when dealing with 5d (and perhaps 4d) transition metals In fact, forthese metals, the local density approximation gives better calculated results (when com-pared to experimental data) than if the gradient corrected approximation is used [53].When studying clusters and molecules, errors can cancel out and one approximationmight be better suited than another Within a certain level of approximation, there arealso several different functionals that one might use, some proving to estimate certainenergies a little better than others [51] The menu at times spoils one for choice andmuch calibration has to be done in order to validate the choice made

1.6 Aims and organization of this work

There are two aims in this work First, I am interested in investigating the suitability

of using graphene as a support material for small Fe, Co and Ni homonuclear clustersand small Fe, Co, Ni and Pt heteronuclear clusters The binding energies of these

1.6 Aims and organization of this work 33

clusters in various adsorbed configurations are calculated to estimate how well theseclusters bind to graphene and to understand what factors contribute to the stability ofthe adsorbed cluster Further to this and perhaps more importantly, the projected andtotal magnetic moments of the adsorbed clusters are calculated and compared to those

of the free clusters In light of the latter, enhancement or retainment of the magneticmoment when bound to graphene does point to the suitability of using graphene inthe development of novel magnetic materials Second, it is important to question ifgraphene might allow for the magnetic moments, or at the least the projected moments

of the atoms in the cluster, to be enhanced based on the interactions with graphene It istherefore important to consider how a mixed cluster, based on the conclusions arrived

at from the first aim of this work, would allow for both enhanced binding as well asenhanced magnetic moments

This work is organized as follows In chapter 2, I will review the basic theory that isinvolved in carrying out the calculations in this work While much of what is found inchapter 2 may be found in standard quantum physics or quantum chemistry textbooks,

I have chosen to review its contents mainly for my own purpose but also to present

a reader who is unfamiliar with quantum mechanical based calculations the necessaryknowledge that lends support to the results that follow As no results are presented

in chapter 2, the reader may choose to skip this chapter In chapter 3, I will presentthe results of my calculations on homonuclear Fe, Co and Ni adatoms and dimers ongraphene This includes a discussion on the adatoms and dimers binding site adsorptionenergies, and projected electronic configurations and magnetic moments I will also dis-cuss the band structures and density of states for the adatom-graphene systems For theadsorbed dimers, two configurations that have not been previously considered in previ-ous studies are presented The gaseous and adsorbed states for each of the adatoms anddimers are compared in the context of understanding the interaction of graphene withthese clusters and its suitability as a support material In chapter 4, I go on to discussthe heteronuclear dimers Aside from mixing the ferromagnetic metals Fe, Co and Ni,

it would be interesting to consider how a metal that is nonmagnetic in the bulk phase

1.6 Aims and organization of this work 34

affects the projected magnetic moments when mixed with the aforementioned metals

in a dimer In this work, I have chosen to consider Pt given the fact that this material

is often mixed with Fe and/or Co in data storage devices or recording media fore, the FeCo, FeNi, CoNi, FePt, CoPt and NiPt mixed dimers are studied in both thegaseous and adsorbed states Again, the adsorption site binding energies, and projectedelectronic configurations and magnetic moments are calculated The aim in this chapter

There-is to investigate magnetic enhancement and reduction in heteronuclear dimers and howthese are affected when the dimers are adsorbed on graphene A further question to beraised is if these dimers can be configured to bind strongly to graphene while at the sametime allowing for further magnetic enhancement Some of the conclusions derived fromthe calculations presented in Chapter 3 will be used in setting up the experiments in thischapter Based on the results of both Chapter 3 and Chapter 4, the free and adsorbedhomonuclear and heteronuclear trimers and tetramers (Fe, Co, Ni and Pt) are studiedand the results of these calculations are presented in Chapter 5 Particular attention

is paid to the projected magnetic moments of the adsorbed clusters and if mixing ofthese clusters can result in a net increase in the total magnetic moment of the adsorbedcluster, and if mixing is energetically favorable when adsorbed on graphene I pointout to the reader here that the kinetics of mixing is not studied in this work Finally, Iwill conclude this thesis in Chapter 6 by summarizing the key findings from the resultschapters, as well as discuss further work that can be done to progress our understanding

of the systems studied here

2.1 The Schr¨odinger equation and Dirac notation

I begin with the usual introduction to the quantum mechanics: the Schr¨odinger tion The non-relativisitic, time-independent Schr¨odinger equation is written as follows:

equa-ˆ

35

2.1 Schr¨odinger equation and Dirac notation 36

where ψ is a function of spatial and spin coordinates Every physical aspect associatedwith a system described by ψ is contained within the wavefunction itself, which is alsooften referred to as the eigenstate of the system of interest The Hamiltonian ˆHoperates

on this eigenstate in order to ’extract’ the eigenvalues of energy Expressed in full, theabove equation then becomes:

+ 12

or a linear vector space, implies that the wavefunction is square integrable These tworequirements are essential in the quantum mechanics

Dirac devised a beautiful notation for expressing functions that belong to a linearvector space: any complex function is denoted by the ket |ψi and its complex conju-gate denoted by the bra hψ| The beautiful aspect here is that all the axioms involved

in the mathematical description of linear vector spaces are succinctly encapsulated inhow these kets and bras operate In fact one need not know anything about linear vectorspaces and still apply this notation with success Particularly useful is the physical real-ity associated with the ket and the bra, especially in context of the quantum mechanics

It is fitting that I delve a little into what the bra and ket represent so as to relate thereader to physical meaning rather than just exploring a bunch of equations

The ket |ψi can be thought of as representing a ‘present’ state of the system (e.g

a system of interacting electrons and nuclei) and the bra hψ| as representing a ‘future’

2.2 The Variational Principle 37

state (e.g the final state of the system of electrons upon interacting with or perturbed

by the electromagnetic field of an incident photon) The inner product of the bra and ketthen gives the probability amplitude associated with collapsing the initial state to someknown final state For example, an electron in a box with state |ψi has a probability am-plitude hx|ψi of being found at some point x within the box This can also be expressed

in function form as follows: hx|ψi = ψ(x) I will make use of both notations in thecourse of this thesis The probability amplitude, ψ (x), has no real physical or tangiblemeaning Rather, it is the probability itself that allows us to relate or make sense ofthe system of interest The probability function is given by the square modulus of thewavefunction, viz |ψ (x)|2 is the probability that a particle described by the eigenstate

ψ can be found at position x

2.2 The Variational Principle

In order to determine the spectrum of energy eigenvalues, we require knowledge

of both the eigenstate and Hamiltonian of the system The idea here is to guess at theeigenstate of the system, using basis functions to aid this guess, and to then minimizethe total energy of the system of electrons in order to obtain its true ground state energy(or as close to it as possible, limited by the various approximations made in the calcu-lations) As with any minimization process, we require that δE = 0 The variationalprinciple provides proof that this minimization process is equivalent to the Schr¨odingerequation, ˆHψ = Eψ since any guess at the eigenstate of the system would give aneigenvalue that is greater than or equal to the true ground state energy

Theorem 1 (The Variation Theorem) Given a system whose Hamiltonian ˆH is timeindependent and whose lowest energy eigenvalue is E0, and anyψ that is a normalizedand well-behaved function of the coordinates of the particles in the system that satisfiesthe necessary boundary conditions, thenR ψ∗Hψdx ≥ Eˆ 0

PROOF The eigenfunction of the system can be expressed as a linear combination ofbasis functions, φ, as follows: ψ= P aiφi, where ai = hφi|ψi Substituting this into the

2.2 The Variational Principle 38

Schr¨odinger equation, ˆHψ = Eψ, we get:

Z

ψ∗Hˆψ =

Z X

We defined E0as the lowest energy eigenvalue in the spectrum of eigenvalues obtained

by solving the Schr¨odinger equation Therefore, Ek ≥ E0 This implies:

δE ψ hψ|ψi + E ψ {hδψ|ψi + hψ|δψi} = hδψ| ˆH |ψi + hψ| ˆH |δψi

δE ψ = hδψ| ˆH − E |ψi + hψ| ˆH − E |δψi (2.4)

2.3 The Hellmann-Feynman Theorem 39

Equation 2.4 implies that if the energy is to be a minimum, then hδψ| ˆH − E |ψi +hψ| ˆH −

E |δψi = 0 Given the fact that ˆH is Hermitian (i.e hψ| ˆHψi = hψ ˆH|ψi), and by letting

min-2.3 The Hellmann-Feynman Theorem

Whilst the variational and minimum principles allow us to obtain an electronic ergy minimum for a given system of atoms (i.e for a particular set of atomic coor-dinates), we are usually interested in optimizing the geometry of this system To dothis we need to calculate, at the end of each electronic energy minimization process,the forces that act on the nuclei in order for the system to ‘slide down’ its potentialenergy surface These forces are determined via application of the Hellmann-Feynmantheorem:

Theorem 3 (The Hellmann-Feynman Theorem) The variation of the electronic ergy as a function ofλ depends on the variation of the related Hamiltonian with respect

en-toλ itself

PROOF Given an orthonormalized total wavefunction of a system ψ, we have the